Figure 2.125 shows a uniform rigid bar of mass \(m\) and length \(l\), pivoted at one end (point \(O\) ) and carrying a circular disk of mass \(M\) and mass moment of inertia \(J\) (about its rotational axis) at the other end (point \(P\) ). The circular disk is connected to a spring of stiffness \(k\) and a viscous damper of damping constant \(c\) as indicated.

a. Derive the equation of motion of the system for small angular displacements of the rigid bar about the pivot point \(O\) and express it in the form:
\[m_{0} \ddot{\theta}+c_{0} \dot{\theta}+k_{0} \theta=0\]

b. Derive conditions corresponding to the stable, unstable, and marginally stable behavior of the system.

Transcribed Image Text:

00000 k P. L H C FIGURE 2.125 Angular displacement of a rigid bar.